It is an example of a extension and reduction of a principal bundle.
Suppose $P$ is the frame bundle $FM$ of a manifold $M$. In this case, $G=GL(n)$, and consider $H=O(n)$. The orbit space $P/H=FM/O(n)$ is, I guess, also a bundle over $M$. The fibre $(P/H)_x$ consists of classes of basis of $T_x M$ where the equivalence of basis is given by the existence of an orthogonal transformation between them.
Now, if we have a section $\sigma$ of the bundle $P/H$ we are assigning in a smooth way a class of bases of $T_x M$ for every $x\in M$. We have, then, a new bundle principal bundle $Q=\bigsqcup\limits_{x\in M} \{x\}\times \sigma(x)$ with group $O(n)$. It can be shown that is a reduction of the frame bundle.
On the other hand, every class $\sigma(x)$ determines a bilinear form in $T_x M$, indeed, a metric (see this). So we have a Riemannian metric on $M$. On the contrary, a Riemannian metric $g$ let us specify a section $\sigma$ of $P/H$.
By the way, as we know (see here), every manifold can be equipped with such a metric, and so we always have the required section.
This example is a particular case of a G-structure.
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Author of the notes: Antonio J. Pan-Collantes
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